Summary: PolyGamma Functions of Negative Order
VICTOR S. ADAMCHIK
January 22, 1998
Abstract
Liouville's fractional integration is used to de ne polygamma func-
tions n
z for negative integer n. It's shown that such n
z can
be represented in a closed form by means of the rst derivatives of
the Hurwitz Zeta function. Relations to the Barnes G-function and
generalized Glaisher's constants are also discussed.
1 Introduction
The idea to de ne the polygamma function
z for every complex via
Liouville's fractional integration operator is quite natural and was around
for a while see Ross 1974 and Grossman 1976. However, for arbitrary
negative integer the closed form of
z was not developed yet - the
only two particular cases = ,2 and = ,3 have been studied see Gosper
1997. It is the purpose of this note is to consider